This is a basic explanation of what happens.
Say bob is watching me fall into a black hole.
bob sees things quite differently from me. As I get closer and closer to the horizon, he sees me move more and more slowly. In fact, no matter how long he waits, he will never quite see me reach the horizon.

Why does he see things this way? The best way to think about it is that it`s really just an optical illusion. It doesn`t really take an infinite amount of time for you to cross the horizon. As you get closer and closer to the horizon, the light that you`re emitting takes longer and longer to climb back out to reach bob. In fact, the radiation you emit right as you cross the horizon will hover right there at the horizon forever and never reach him. You`ve long since passed through the horizon, but the light signal telling him that won`t reach him for an infinitely long time.